Armors, custom work, and pricing:
I do know a bit about making armor (more from an academic than a practical standpoint, anyway), and can reccommend a couple of books on the subject, if you're interested... Anyway, YOU are right that the Halfling armor would cost as much as normal armor, ordinarily. YOUR PLAYER is correct that, in Medieval times, all crafts are basically custom made... Remember, there are no machinery-made mass-produced items. Everything is made by hand.
Are there items sitting around on the shelves? Sure there are! What do you think the smith is doing with his time? Whenever there are no customers in the shop, he will be practicing his art, making something. How much stuff does he have? Depends on the area... If armors are regularly being sold to a large adventuring population, he won't have time to make enough (and more smiths will be opening shops). If it is an area with only occasional sales, he will have as many items sitting around as he has room to store (and may lower his prices).
Even with all items being hand-made and basically custom, however, it shouldn't take additional time to make most forms of armor... Making Halfling maille is no harder than Human maille. The rings are the same size, and the suit is smaller. The exceptions are armors which require "forms".
When the cuirboilli breastplate for Leather Armor is made, for instance, leather is wrapped around a wooden form, which is then immersed in a pot of boiling parafin (wax). The boiling process causes it to shrink, and become almost as hard as metal. Now MAKING the armor is no more difficult, but if you don't have the form for a child/Halfling, then you have to have a woodcarver make one...
Any plate (platemail, field plate, full plate), and even breastplates, banded, lamellar, splint, etc., will have to be forged on an anvil, and probably uses some sort of form, as well. Since the smith doesn't really need the form, except for measuring, however, (s)he could still do the work without it, PROVIDING that the Halfling made themself available for frequent "fittings" to make sure the armor fit right... OR, they could pay to have Halfling-sized forms made to their measurements, so that the smith could use them, instead (so there's your excuse for charging a bit extra, but personally, I'd limit it to - say - 10% more than the base price).
By the bayou, howsomever, the idea that a Halfling-sized suit of maille (improperly called "chainmail", a Victorian-era misonym for maille) would cost as much as a normal-sized one is ridiculous. It would actually take both less time and materials to make the smaller shirt. Smiths pre-made rings of maille, and had them sitting around, waiting for repairs or the making of new suits, so the PHB prices (while official!) are "wrong", in my eyes... Then again, find me a Fantasy RPG whose prices AREN'T, eh?
The entire Crafts system needs an overhaul... For a craftsman capable of making both, an item with a DC of five will take MORE time than an item with a DC of 20! There is something seriously wrong with that, IMHO! YMMV.
I do know a bit about making armor (more from an academic than a practical standpoint, anyway), and can reccommend a couple of books on the subject, if you're interested... Anyway, YOU are right that the Halfling armor would cost as much as normal armor, ordinarily. YOUR PLAYER is correct that, in Medieval times, all crafts are basically custom made... Remember, there are no machinery-made mass-produced items. Everything is made by hand.
Are there items sitting around on the shelves? Sure there are! What do you think the smith is doing with his time? Whenever there are no customers in the shop, he will be practicing his art, making something. How much stuff does he have? Depends on the area... If armors are regularly being sold to a large adventuring population, he won't have time to make enough (and more smiths will be opening shops). If it is an area with only occasional sales, he will have as many items sitting around as he has room to store (and may lower his prices).
Even with all items being hand-made and basically custom, however, it shouldn't take additional time to make most forms of armor... Making Halfling maille is no harder than Human maille. The rings are the same size, and the suit is smaller. The exceptions are armors which require "forms".
When the cuirboilli breastplate for Leather Armor is made, for instance, leather is wrapped around a wooden form, which is then immersed in a pot of boiling parafin (wax). The boiling process causes it to shrink, and become almost as hard as metal. Now MAKING the armor is no more difficult, but if you don't have the form for a child/Halfling, then you have to have a woodcarver make one...
Any plate (platemail, field plate, full plate), and even breastplates, banded, lamellar, splint, etc., will have to be forged on an anvil, and probably uses some sort of form, as well. Since the smith doesn't really need the form, except for measuring, however, (s)he could still do the work without it, PROVIDING that the Halfling made themself available for frequent "fittings" to make sure the armor fit right... OR, they could pay to have Halfling-sized forms made to their measurements, so that the smith could use them, instead (so there's your excuse for charging a bit extra, but personally, I'd limit it to - say - 10% more than the base price).
By the bayou, howsomever, the idea that a Halfling-sized suit of maille (improperly called "chainmail", a Victorian-era misonym for maille) would cost as much as a normal-sized one is ridiculous. It would actually take both less time and materials to make the smaller shirt. Smiths pre-made rings of maille, and had them sitting around, waiting for repairs or the making of new suits, so the PHB prices (while official!) are "wrong", in my eyes... Then again, find me a Fantasy RPG whose prices AREN'T, eh?

The entire Crafts system needs an overhaul... For a craftsman capable of making both, an item with a DC of five will take MORE time than an item with a DC of 20! There is something seriously wrong with that, IMHO! YMMV.